Abstract:We encode arbitrary finite impartial combinatorial games in terms
of lattice points in rational convex polyhedra. Encodings provided
by these lattice games can be made particularly efficient for octal
games, which we generalize to squarefree games. These encompass
all heap games in a natural setting where the Sprague­Grundy
theorem for normal play manifests itself geometrically. We provide
an algorithm to compute normal play strategies.
The setting of lattice games naturally allows for misère play, where
0 is declared a losing position. Lattice games also allow situations
where larger finite sets of positions are declared losing. Generating
functions for sets of winning positions provide data structures for
strategies of lattice games. We conjecture that every lattice game
has a rational strategy: a rational generating function for its winning
positions. Additionally, we conjecture that every lattice game has
an affine stratification: a partition of its set of winning positions
into a finite disjoint union of finitely generated modules for affine
semigroups. This conjecture is true for normal-play squarefree
games and every lattice game with finite misère quotient.